Physlib.Relativity.PauliMatrices.ToTensor

Imports

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
module

public import Physlib.Relativity.Tensors.ComplexTensor.Metrics.Basic
public import Physlib.Relativity.PauliMatrices.AsTensor
public import Physlib.Relativity.Tensors.ComplexTensor.Metrics.Basic

Pauli matrices as a tensor

@[expose] public section

Tensorial structure

The tensorial structure on the type Fin 1 ⊕ Fin 3 → Matrix (Fin 2) (Fin 2) ℂ and properties thereof.

The equivalence between the type of indices of a [.up, .upL, .upR] tensor and (Fin 1 ⊕ Fin 3) × Fin 2 × Fin 2.

def indexEquiv : ComponentIdx (S := complexLorentzTensor) ![.up, .upL, .upR] ≃
    (Fin 1 ⊕ Fin 3) × Fin 2 × Fin 2 where
  toFun v := (finSumFinEquiv.symm (v 0 : Fin 4), v 1, v 2)
  invFun v := fun | 0 => finSumFinEquiv v.1 | 1 => v.2.1 | 2 => v.2.2
  left_inv v := byv:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (fun v x =>
      match x with
      | 0 => finSumFinEquiv v.1
      | 1 => v.2.1
      | 2 => v.2.2)
    ((fun v => (finSumFinEquiv.symm (v 0), v 1, v 2)) v) =
  v
    funext xv:ComponentIdx ![Color.up, Color.upL, Color.upR]x:Fin (Nat.succ 0).succ.succ⊢ (fun v x =>
      match x with
      | 0 => finSumFinEquiv v.1
      | 1 => v.2.1
      | 2 => v.2.2)
    ((fun v => (finSumFinEquiv.symm (v 0), v 1, v 2)) v) x =
  v x
    simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, Equiv.apply_symm_apply]v:ComponentIdx ![Color.up, Color.upL, Color.upR]x:Fin (Nat.succ 0).succ.succ⊢ (match x with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v x
    fin_cases x«0»v:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (match (fun i => i) ⟨0, ⋯⟩ with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v ((fun i => i) ⟨0, ⋯⟩)«1»v:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (match (fun i => i) ⟨1, ⋯⟩ with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v ((fun i => i) ⟨1, ⋯⟩)«2»v:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (match (fun i => i) ⟨2, ⋯⟩ with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v ((fun i => i) ⟨2, ⋯⟩)
    <;>«0»v:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (match (fun i => i) ⟨0, ⋯⟩ with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v ((fun i => i) ⟨0, ⋯⟩)«1»v:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (match (fun i => i) ⟨1, ⋯⟩ with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v ((fun i => i) ⟨1, ⋯⟩)«2»v:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ (match (fun i => i) ⟨2, ⋯⟩ with
  | 0 => v 0
  | 1 => v 1
  | 2 => v 2) =
  v ((fun i => i) ⟨2, ⋯⟩) rflAll goals completed! 🐙
  right_inv v := byv:(Fin 1 ⊕ Fin 3) × Fin 2 × Fin 2⊢ (fun v => (finSumFinEquiv.symm (v 0), v 1, v 2))
    ((fun v x =>
        match x with
        | 0 => finSumFinEquiv v.1
        | 1 => v.2.1
        | 2 => v.2.2)
      v) =
  v
    simpAll goals completed! 🐙

              instance tensorial : TensorSpecies.Tensorial complexLorentzTensor
    ![.up, .upL, .upR] (Fin 1 ⊕ Fin 3 → Matrix (Fin 2) (Fin 2) ℂ) where
  toTensor := LinearEquiv.symm <|
    Equiv.toLinearEquiv
    ((Tensor.basis (S := complexLorentzTensor) ![.up, .upL, .upR]).repr.toEquiv.trans <|
  Finsupp.equivFunOnFinite.trans <|
  (Equiv.piCongrLeft' _ indexEquiv).trans <|
  (Equiv.curry _ _ _).trans <|
  Equiv.piCongrRight fun _ => Equiv.curry _ _ _)
    { map_add := fun x y => byx:complexLorentzTensor.Tensor ![Color.up, Color.upL, Color.upR]y:complexLorentzTensor.Tensor ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr.toEquiv.trans
      (Finsupp.equivFunOnFinite.trans
        ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv).trans
          ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ).trans
            (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)))))
    (x + y) =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr.toEquiv.trans
        (Finsupp.equivFunOnFinite.trans
          ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv).trans
            ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ).trans
              (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)))))
      x +
    ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr.toEquiv.trans
        (Finsupp.equivFunOnFinite.trans
          ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv).trans
            ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ).trans
              (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)))))
      y
        simp [Nat.succ_eq_add_one, Nat.reduceAdd, map_add]x:complexLorentzTensor.Tensor ![Color.up, Color.upL, Color.upR]y:complexLorentzTensor.Tensor ![Color.up, Color.upL, Color.upR]⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    (Function.curry
      ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
        (Finsupp.equivFunOnFinite
          ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr x +
            (Tensor.basis ![Color.up, Color.upL, Color.upR]).repr y)))) =
  (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
      (Function.curry
        ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
          (Finsupp.equivFunOnFinite ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr x)))) +
    (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
      (Function.curry
        ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
          (Finsupp.equivFunOnFinite ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr y))))
        rflAll goals completed! 🐙
      map_smul := fun c x => byc:ℂx:complexLorentzTensor.Tensor ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr.toEquiv.trans
      (Finsupp.equivFunOnFinite.trans
        ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv).trans
          ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ).trans
            (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)))))
    (c • x) =
  c •
    ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr.toEquiv.trans
        (Finsupp.equivFunOnFinite.trans
          ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv).trans
            ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ).trans
              (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)))))
      x
        simp [Nat.succ_eq_add_one, Nat.reduceAdd, _root_.map_smul]c:ℂx:complexLorentzTensor.Tensor ![Color.up, Color.upL, Color.upR]⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    (Function.curry
      ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
        (Finsupp.equivFunOnFinite (c • (Tensor.basis ![Color.up, Color.upL, Color.upR]).repr x)))) =
  c •
    (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
      (Function.curry
        ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
          (Finsupp.equivFunOnFinite ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr x))))
        rflAll goals completed! 🐙}lemma toTensor_symm_apply (p : ℂT[.up, .upL, .upR]) :
    (toTensor (self := tensorial)).symm p =
    ((Equiv.piCongrRight fun _ => Equiv.curry _ _ _) <|
    (Equiv.curry _ _ _) <|
    Equiv.piCongrLeft' _ indexEquiv <|
    Finsupp.equivFunOnFinite <|
    (Tensor.basis (S := complexLorentzTensor) _).repr p) := rfl

lemma toTensor_symm_basis (b : (x : Fin (Nat.succ 0).succ.succ) →
    Fin (complexLorentzTensor.repDim (![Color.up, Color.upL, Color.upR] x))) :
    (toTensor (self := tensorial)).symm (Tensor.basis ![Color.up, Color.upL, Color.upR] b) =
    fun μ α β => if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0 := byb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))⊢ toTensor.symm ((Tensor.basis ![Color.up, Color.upL, Color.upR]) b) = fun μ α β =>
  if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0
  rw [toTensor_symm_applyb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ)
      ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
        (Finsupp.equivFunOnFinite
          ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
            ((Tensor.basis ![Color.up, Color.upL, Color.upR]) b))))) =
  fun μ α β => if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ)
      ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
        (Finsupp.equivFunOnFinite
          ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
            ((Tensor.basis ![Color.up, Color.upL, Color.upR]) b))))) =
  fun μ α β => if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0]b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    ((Equiv.curry (Fin 1 ⊕ Fin 3) (Fin 2 × Fin 2) ℂ)
      ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv)
        (Finsupp.equivFunOnFinite
          ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
            ((Tensor.basis ![Color.up, Color.upL, Color.upR]) b))))) =
  fun μ α β => if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0
  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Basis.repr_self, Finsupp.equivFunOnFinite_single,
    Equiv.curry_apply, Fin.isValue, cons_val_one, cons_val_zero, cons_val]b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    (Function.curry ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv) (Pi.single b 1))) =
  fun μ α β => if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0
  funext μ α βb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (Equiv.piCongrRight fun x => Equiv.curry (Fin 2) (Fin 2) ℂ)
    (Function.curry ((Equiv.piCongrLeft' (fun a => ℂ) indexEquiv) (Pi.single b 1))) μ α β =
  if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0
  simp only [Equiv.piCongrRight_apply, Equiv.curry_apply, Pi.map_apply, Function.curry_apply,
    Equiv.piCongrLeft'_apply, Fin.isValue]b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ Pi.single b 1 (indexEquiv.symm (μ, α, β)) = if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0
  rw [Pi.single_applyb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (if indexEquiv.symm (μ, α, β) = b then 1 else 0) = if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (if indexEquiv.symm (μ, α, β) = b then 1 else 0) = if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0]b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (if indexEquiv.symm (μ, α, β) = b then 1 else 0) = if b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β then 1 else 0
  congre_cb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (indexEquiv.symm (μ, α, β) = b) = (b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β)
  simp [indexEquiv]e_cb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (fun x =>
      match x with
      | 0 => finSumFinEquiv μ
      | 1 => α
      | 2 => β) =
    b ↔
  b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β
  constructore_c.mpb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (fun x =>
      match x with
      | 0 => finSumFinEquiv μ
      | 1 => α
      | 2 => β) =
    b →
  b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = βe_c.mprb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β →
  (fun x =>
      match x with
      | 0 => finSumFinEquiv μ
      | 1 => α
      | 2 => β) =
    b
  ·e_c.mpb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (fun x =>
      match x with
      | 0 => finSumFinEquiv μ
      | 1 => α
      | 2 => β) =
    b →
  b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β intro he_c.mpb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:(fun x =>
    match x with
    | 0 => finSumFinEquiv μ
    | 1 => α
    | 2 => β) =
  b⊢ b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β
    subst he_c.mpμ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ (fun x =>
        match x with
        | 0 => finSumFinEquiv μ
        | 1 => α
        | 2 => β)
      0 =
    finSumFinEquiv μ ∧
  (fun x =>
          match x with
          | 0 => finSumFinEquiv μ
          | 1 => α
          | 2 => β)
        1 =
      α ∧
    (fun x =>
          match x with
          | 0 => finSumFinEquiv μ
          | 1 => α
          | 2 => β)
        2 =
      β
    simpAll goals completed! 🐙
  ·e_c.mprb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β →
  (fun x =>
      match x with
      | 0 => finSumFinEquiv μ
      | 1 => α
      | 2 => β) =
    b intro he_c.mprb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (fun x =>
    match x with
    | 0 => finSumFinEquiv μ
    | 1 => α
    | 2 => β) =
  b
    funext xe_c.mprb:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = βx:Fin 3⊢ (match x with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b x
    fin_cases xe_c.mpr.«0»b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (match (fun i => i) ⟨0, ⋯⟩ with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b ((fun i => i) ⟨0, ⋯⟩)e_c.mpr.«1»b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (match (fun i => i) ⟨1, ⋯⟩ with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b ((fun i => i) ⟨1, ⋯⟩)e_c.mpr.«2»b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (match (fun i => i) ⟨2, ⋯⟩ with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b ((fun i => i) ⟨2, ⋯⟩)
    ·e_c.mpr.«0»b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (match (fun i => i) ⟨0, ⋯⟩ with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b ((fun i => i) ⟨0, ⋯⟩) simp [h.1]All goals completed! 🐙
    ·e_c.mpr.«1»b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (match (fun i => i) ⟨1, ⋯⟩ with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b ((fun i => i) ⟨1, ⋯⟩) simp [h.2.1]All goals completed! 🐙
    ·e_c.mpr.«2»b:(x : Fin (Nat.succ 0).succ.succ) → Fin (repDim (![Color.up, Color.upL, Color.upR] x))μ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2h:b 0 = finSumFinEquiv μ ∧ b 1 = α ∧ b 2 = β⊢ (match (fun i => i) ⟨2, ⋯⟩ with
  | 0 => finSumFinEquiv μ
  | 1 => α
  | 2 => β) =
  b ((fun i => i) ⟨2, ⋯⟩) simp [h.2.2]All goals completed! 🐙

Pauli matrices as a tensor

The Pauli matrices as a tensor toTensor pauliMatrix in ℂT[.up, .upL, .upR].

scoped[PauliMatrix] notation "σ^^^" => toTensor pauliMatrix

              

set_option backward.isDefEq.respectTransparency false in
lemma toTensor_basis_expand : σ^^^ =
    Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (0 : Fin 4) | 1 => (0 : Fin 2) | 2 => (0 : Fin 2))
    + Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (0 : Fin 4) | 1 => (1 : Fin 2) | 2 => (1 : Fin 2))
    + Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (1 : Fin 4) | 1 => (0 : Fin 2) | 2 => (1 : Fin 2))
    + Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (1 : Fin 4) | 1 => (1 : Fin 2) | 2 => (0 : Fin 2))
    - I • Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (2 : Fin 4) | 1 => (0 : Fin 2) | 2 => (1 : Fin 2))
    + I • Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (2 : Fin 4) | 1 => (1 : Fin 2) | 2 => (0 : Fin 2))
    + Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (3 : Fin 4) | 1 => (0 : Fin 2) | 2 => (0 : Fin 2))
    - Tensor.basis ![Color.up, Color.upL, Color.upR]
      (fun | 0 => (3 : Fin 4) | 1 => (1 : Fin 2) | 2 => (1 : Fin 2)) := by⊢ toTensor σ =
  ((((((((Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 0
                  | 1 => 0
                  | 2 => 0) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 0
                  | 1 => 1
                  | 2 => 1) +
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 1
                | 1 => 0
                | 2 => 1) +
            (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
              match x with
              | 0 => 1
              | 1 => 1
              | 2 => 0) -
          I •
            (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
              match x with
              | 0 => 2
              | 1 => 0
              | 2 => 1) +
        I •
          (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
            match x with
            | 0 => 2
            | 1 => 1
            | 2 => 0) +
      (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
        match x with
        | 0 => 3
        | 1 => 0
        | 2 => 0) -
    (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
      match x with
      | 0 => 3
      | 1 => 1
      | 2 => 1
  apply toTensor (self := tensorial).symm.injective⊢ toTensor.symm (toTensor σ) =
  toTensor.symm
    (((((((((Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 0
                    | 1 => 0
                    | 2 => 0) +
                  (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 0
                    | 1 => 1
                    | 2 => 1) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 1
                  | 1 => 0
                  | 2 => 1) +
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 1
                | 1 => 1
                | 2 => 0) -
            I •
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 2
                | 1 => 0
                | 2 => 1) +
          I •
            (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
              match x with
              | 0 => 2
              | 1 => 1
              | 2 => 0) +
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 0
          | 2 => 0) -
      (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
        match x with
        | 0 => 3
        | 1 => 1
        | 2 => 1)
  simp [toTensor_symm_basis]⊢ σ =
  (((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
            fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
        I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
    fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0
  funext μ α βμ:Fin 1 ⊕ Fin 3α:Fin 2β:Fin 2⊢ σ μ α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    μ α β
  fin_cases μ«0»α:Fin 2β:Fin 2⊢ σ (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) α β«1»α:Fin 2β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) α β«2»α:Fin 2β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) α β«3»α:Fin 2β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) α β <;>«0»α:Fin 2β:Fin 2⊢ σ (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) α β«1»α:Fin 2β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) α β«2»α:Fin 2β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) α β«3»α:Fin 2β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) α β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) α β fin_cases α«3».«0»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β«3».«1»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β <;>«0».«0»β:Fin 2⊢ σ (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β«0».«1»β:Fin 2⊢ σ (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β«1».«0»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β«1».«1»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β«2».«0»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β«2».«1»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β«3».«0»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨0, ⋯⟩) β«3».«1»β:Fin 2⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) β fin_cases β«3».«1».«0»⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)«3».«1».«1»⊢ σ (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ((((((((fun μ α β => if 0 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) + fun μ α β =>
                  if 0 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0) +
                fun μ α β => if 1 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
              fun μ α β => if 1 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) -
            I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 0 = α ∧ 1 = β then 1 else 0) +
          I • fun μ α β => if 2 = finSumFinEquiv μ ∧ 1 = α ∧ 0 = β then 1 else 0) +
        fun μ α β => if 3 = finSumFinEquiv μ ∧ 0 = α ∧ 0 = β then 1 else 0) -
      fun μ α β => if 3 = finSumFinEquiv μ ∧ 1 = α ∧ 1 = β then 1 else 0)
    (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)
  all_goals
    simp«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = (if 0 = finSumFinEquiv (Sum.inr 2) then 1 else 0) - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0
  all_goals
    try rw [if_pos (by⊢ 0 = finSumFinEquiv (Sum.inr 2)«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = (if 0 = finSumFinEquiv (Sum.inr 2) then 1 else 0) - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0 decide⊢ 0 = finSumFinEquiv (Sum.inr 2)«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = (if 0 = finSumFinEquiv (Sum.inr 2) then 1 else 0) - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0)]«1».«1».«0»⊢ σ (Sum.inr 0) 1 0 = 1 + if 2 = finSumFinEquiv (Sum.inr 0) then I else 0«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = (if 0 = finSumFinEquiv (Sum.inr 2) then 1 else 0) - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = (if 0 = finSumFinEquiv (Sum.inr 2) then 1 else 0) - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0
    try rw [if_neg (by⊢ ¬0 = finSumFinEquiv (Sum.inr 2)«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0 decideAll goals completed! 🐙«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0)]«3».«1».«0»⊢ σ (Sum.inr 2) 1 0 = 0 + if 2 = finSumFinEquiv (Sum.inr 2) then I else 0«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0
    try rw [if_neg (by⊢ ¬3 = finSumFinEquiv (Sum.inr 2)«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0 decide⊢ ¬3 = finSumFinEquiv (Sum.inr 2)«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0)]«3».«1».«0»⊢ σ (Sum.inr 2) 1 0 = 0 + 0«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - if 3 = finSumFinEquiv (Sum.inr 2) then 1 else 0
    try rw [if_pos (by⊢ 3 = finSumFinEquiv (Sum.inr 2)«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - 1 decideAll goals completed! 🐙«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - 1)]«3».«0».«0»⊢ σ (Sum.inr 2) 0 0 = 0 + 1«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - 1«3».«1».«1»⊢ σ (Sum.inr 2) 1 1 = 0 - 1
    simp [pauliMatrix]All goals completed! 🐙

lemma toTensor_eq_ofRat : σ^^^ = ofRat (fun b =>
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 4 = repDim (![Color.up, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (0 : Fin 4) ∧ b 1 = b 2 then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 4 = repDim (![Color.up, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (1 : Fin 4) ∧ b 1 ≠ b 2 then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 4 = repDim (![Color.up, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨0, -1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 4 = repDim (![Color.up, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨0, 1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 4 = repDim (![Color.up, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 4 = repDim (![Color.up, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ 2 = repDim (![Color.up, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨-1, 0⟩ else 0) := by⊢ toTensor σ =
  ofRat fun b =>
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis ![Color.up, Color.upL, Color.upR]).repr (toTensor σ) =
  (Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
    (ofRat fun b =>
      if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
              else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0)
  ext bb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr (toTensor σ)) b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    b
  rw [toTensor_basis_expandb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (((((((((Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 0
                      | 2 => 0) +
                    (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 1
                      | 2 => 1) +
                  (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 1
                    | 1 => 0
                    | 2 => 1) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 1
                  | 1 => 1
                  | 2 => 0) -
              I •
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 0
                  | 2 => 1) +
            I •
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 2
                | 1 => 1
                | 2 => 0) +
          (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
            match x with
            | 0 => 3
            | 1 => 0
            | 2 => 0) -
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 1
          | 2 => 1))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (((((((((Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 0
                      | 2 => 0) +
                    (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 1
                      | 2 => 1) +
                  (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 1
                    | 1 => 0
                    | 2 => 1) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 1
                  | 1 => 1
                  | 2 => 0) -
              I •
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 0
                  | 2 => 1) +
            I •
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 2
                | 1 => 1
                | 2 => 0) +
          (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
            match x with
            | 0 => 3
            | 1 => 0
            | 2 => 0) -
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 1
          | 2 => 1))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    b]b:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (((((((((Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 0
                      | 2 => 0) +
                    (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 1
                      | 2 => 1) +
                  (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 1
                    | 1 => 0
                    | 2 => 1) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 1
                  | 1 => 1
                  | 2 => 0) -
              I •
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 0
                  | 2 => 1) +
            I •
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 2
                | 1 => 1
                | 2 => 0) +
          (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
            match x with
            | 0 => 3
            | 1 => 0
            | 2 => 0) -
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 1
          | 2 => 1))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    b
  simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, cons_val_zero, cons_val_one]b:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (((((((((Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 0
                      | 2 => 0) +
                    (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 1
                      | 2 => 1) +
                  (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 1
                    | 1 => 0
                    | 2 => 1) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 1
                  | 1 => 1
                  | 2 => 0) -
              I •
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 0
                  | 2 => 1) +
            I •
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 2
                | 1 => 1
                | 2 => 0) +
          (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
            match x with
            | 0 => 3
            | 1 => 0
            | 2 => 0) -
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 1
          | 2 => 1))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    b
  repeat rw [basis_eq_ofRatb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      ((((((((ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 0
                              | 2 => 0) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                    (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                      match x with
                      | 0 => 0
                      | 1 => 1
                      | 2 => 1) +
                  (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                    match x with
                    | 0 => 1
                    | 1 => 0
                    | 2 => 1) +
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 1
                  | 1 => 1
                  | 2 => 0) -
              I •
                (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 0
                  | 2 => 1) +
            I •
              (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
                match x with
                | 0 => 2
                | 1 => 1
                | 2 => 0) +
          (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
            match x with
            | 0 => 3
            | 1 => 0
            | 2 => 0) -
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 1
          | 2 => 1))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      ((((((((ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 0
                              | 2 => 0) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                    ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 1
                              | 2 => 1) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                  ofRat fun b' =>
                    if
                        (fun x =>
                            match x with
                            | 0 => 1
                            | 1 => 0
                            | 2 => 1) =
                          b' then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) +
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 1
                          | 1 => 1
                          | 2 => 0) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) -
              I •
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 2
                          | 1 => 0
                          | 2 => 1) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
            I •
              ofRat fun b' =>
                if
                    (fun x =>
                        match x with
                        | 0 => 2
                        | 1 => 1
                        | 2 => 0) =
                      b' then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
          ofRat fun b' =>
            if
                (fun x =>
                    match x with
                    | 0 => 3
                    | 1 => 0
                    | 2 => 0) =
                  b' then
              { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }) -
        ofRat fun b' =>
          if
              (fun x =>
                  match x with
                  | 0 => 3
                  | 1 => 1
                  | 2 => 1) =
                b' then
            { fst := 1, snd := 0 }
          else { fst := 0, snd := 0 }))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    b]b:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      ((((((((ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 0
                              | 2 => 0) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                    ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 1
                              | 2 => 1) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                  ofRat fun b' =>
                    if
                        (fun x =>
                            match x with
                            | 0 => 1
                            | 1 => 0
                            | 2 => 1) =
                          b' then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) +
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 1
                          | 1 => 1
                          | 2 => 0) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) -
              I •
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 2
                          | 1 => 0
                          | 2 => 1) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
            I •
              ofRat fun b' =>
                if
                    (fun x =>
                        match x with
                        | 0 => 2
                        | 1 => 1
                        | 2 => 0) =
                      b' then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
          ofRat fun b' =>
            if
                (fun x =>
                    match x with
                    | 0 => 3
                    | 1 => 0
                    | 2 => 0) =
                  b' then
              { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }) -
        (Tensor.basis ![Color.up, Color.upL, Color.upR]) fun x =>
          match x with
          | 0 => 3
          | 1 => 1
          | 2 => 1))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      ((((((((ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 0
                              | 2 => 0) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                    ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 1
                              | 2 => 1) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                  ofRat fun b' =>
                    if
                        (fun x =>
                            match x with
                            | 0 => 1
                            | 1 => 0
                            | 2 => 1) =
                          b' then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) +
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 1
                          | 1 => 1
                          | 2 => 0) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) -
              I •
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 2
                          | 1 => 0
                          | 2 => 1) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
            I •
              ofRat fun b' =>
                if
                    (fun x =>
                        match x with
                        | 0 => 2
                        | 1 => 1
                        | 2 => 0) =
                      b' then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
          ofRat fun b' =>
            if
                (fun x =>
                    match x with
                    | 0 => 3
                    | 1 => 0
                    | 2 => 0) =
                  b' then
              { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }) -
        ofRat fun b' =>
          if
              (fun x =>
                  match x with
                  | 0 => 3
                  | 1 => 1
                  | 2 => 1) =
                b' then
            { fst := 1, snd := 0 }
          else { fst := 0, snd := 0 }))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      ((((((((ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 0
                              | 2 => 0) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                    ofRat fun b' =>
                      if
                          (fun x =>
                              match x with
                              | 0 => 0
                              | 1 => 1
                              | 2 => 1) =
                            b' then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) +
                  ofRat fun b' =>
                    if
                        (fun x =>
                            match x with
                            | 0 => 1
                            | 1 => 0
                            | 2 => 1) =
                          b' then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) +
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 1
                          | 1 => 1
                          | 2 => 0) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) -
              I •
                ofRat fun b' =>
                  if
                      (fun x =>
                          match x with
                          | 0 => 2
                          | 1 => 0
                          | 2 => 1) =
                        b' then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
            I •
              ofRat fun b' =>
                if
                    (fun x =>
                        match x with
                        | 0 => 2
                        | 1 => 1
                        | 2 => 0) =
                      b' then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
          ofRat fun b' =>
            if
                (fun x =>
                    match x with
                    | 0 => 3
                    | 1 => 0
                    | 2 => 0) =
                  b' then
              { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }) -
        ofRat fun b' =>
          if
              (fun x =>
                  match x with
                  | 0 => 3
                  | 1 => 1
                  | 2 => 1) =
                b' then
            { fst := 1, snd := 0 }
          else { fst := 0, snd := 0 }))
    b =
  ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0))
    b
  simp only [Fin.isValue, map_sub, map_add, _root_.map_smul, Finsupp.coe_sub, Finsupp.coe_add,
    Finsupp.coe_smul, Pi.sub_apply, Pi.add_apply, ofRat_basis_repr_apply, Pi.smul_apply,
    smul_eq_mul, Physlib.RatComplexNum.I_mul_toComplexNum, mul_ite, ne_eq, cons_val_two,
    Nat.succ_eq_add_one, Nat.reduceAdd]b:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
                  (if
                      (fun x =>
                          match x with
                          | 0 => 0
                          | 1 => 0
                          | 2 => 0) =
                        b then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
                Physlib.RatComplexNum.toComplexNum
                  (if
                      (fun x =>
                          match x with
                          | 0 => 0
                          | 1 => 1
                          | 2 => 1) =
                        b then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
              Physlib.RatComplexNum.toComplexNum
                (if
                    (fun x =>
                        match x with
                        | 0 => 1
                        | 1 => 0
                        | 2 => 1) =
                      b then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
            Physlib.RatComplexNum.toComplexNum
              (if
                  (fun x =>
                      match x with
                      | 0 => 1
                      | 1 => 1
                      | 2 => 0) =
                    b then
                { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }) -
          Physlib.RatComplexNum.toComplexNum
            (if
                (fun x =>
                    match x with
                    | 0 => 2
                    | 1 => 0
                    | 2 => 1) =
                  b then
              { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
            else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
        Physlib.RatComplexNum.toComplexNum
          (if
              (fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 1
                  | 2 => 0) =
                b then
            { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
          else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
      Physlib.RatComplexNum.toComplexNum
        (if
            (fun x =>
                match x with
                | 0 => 3
                | 1 => 0
                | 2 => 0) =
              b then
          { fst := 1, snd := 0 }
        else { fst := 0, snd := 0 }) -
    Physlib.RatComplexNum.toComplexNum
      (if
          (fun x =>
              match x with
              | 0 => 3
              | 1 => 1
              | 2 => 1) =
            b then
        { fst := 1, snd := 0 }
      else { fst := 0, snd := 0 }) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ ¬b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0)
  simp only [Fin.isValue, ← map_add, ← map_sub]b:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    ((((((((if
                      (fun x =>
                          match x with
                          | 0 => 0
                          | 1 => 0
                          | 2 => 0) =
                        b then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
                  if
                      (fun x =>
                          match x with
                          | 0 => 0
                          | 1 => 1
                          | 2 => 1) =
                        b then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
                if
                    (fun x =>
                        match x with
                        | 0 => 1
                        | 1 => 0
                        | 2 => 1) =
                      b then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
              if
                  (fun x =>
                      match x with
                      | 0 => 1
                      | 1 => 1
                      | 2 => 0) =
                    b then
                { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }) -
            if
                (fun x =>
                    match x with
                    | 0 => 2
                    | 1 => 0
                    | 2 => 1) =
                  b then
              { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
            else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
          if
              (fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 1
                  | 2 => 0) =
                b then
            { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
          else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
        if
            (fun x =>
                match x with
                | 0 => 3
                | 1 => 0
                | 2 => 0) =
              b then
          { fst := 1, snd := 0 }
        else { fst := 0, snd := 0 }) -
      if
          (fun x =>
              match x with
              | 0 => 3
              | 1 => 1
              | 2 => 1) =
            b then
        { fst := 1, snd := 0 }
      else { fst := 0, snd := 0 }) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ ¬b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0)
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx ![Color.up, Color.upL, Color.upR]⊢ ((((((((if
                    (fun x =>
                        match x with
                        | 0 => 0
                        | 1 => 0
                        | 2 => 0) =
                      b then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
                if
                    (fun x =>
                        match x with
                        | 0 => 0
                        | 1 => 1
                        | 2 => 1) =
                      b then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
              if
                  (fun x =>
                      match x with
                      | 0 => 1
                      | 1 => 0
                      | 2 => 1) =
                    b then
                { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }) +
            if
                (fun x =>
                    match x with
                    | 0 => 1
                    | 1 => 1
                    | 2 => 0) =
                  b then
              { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }) -
          if
              (fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 0
                  | 2 => 1) =
                b then
            { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
          else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
        if
            (fun x =>
                match x with
                | 0 => 2
                | 1 => 1
                | 2 => 0) =
              b then
          { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
        else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
      if
          (fun x =>
              match x with
              | 0 => 3
              | 1 => 0
              | 2 => 0) =
            b then
        { fst := 1, snd := 0 }
      else { fst := 0, snd := 0 }) -
    if
        (fun x =>
            match x with
            | 0 => 3
            | 1 => 1
            | 2 => 1) =
          b then
      { fst := 1, snd := 0 }
    else { fst := 0, snd := 0 }) =
  if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
  else
    if b 0 = Fin.cast ⋯ 1 ∧ ¬b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
          else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0
  revert b⊢ ∀ (b : ComponentIdx ![Color.up, Color.upL, Color.upR]),
  ((((((((if
                      (fun x =>
                          match x with
                          | 0 => 0
                          | 1 => 0
                          | 2 => 0) =
                        b then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
                  if
                      (fun x =>
                          match x with
                          | 0 => 0
                          | 1 => 1
                          | 2 => 1) =
                        b then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) +
                if
                    (fun x =>
                        match x with
                        | 0 => 1
                        | 1 => 0
                        | 2 => 1) =
                      b then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) +
              if
                  (fun x =>
                      match x with
                      | 0 => 1
                      | 1 => 1
                      | 2 => 0) =
                    b then
                { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }) -
            if
                (fun x =>
                    match x with
                    | 0 => 2
                    | 1 => 0
                    | 2 => 1) =
                  b then
              { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
            else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
          if
              (fun x =>
                  match x with
                  | 0 => 2
                  | 1 => 1
                  | 2 => 0) =
                b then
            { fst := 0, snd := 1 } * { fst := 1, snd := 0 }
          else { fst := 0, snd := 1 } * { fst := 0, snd := 0 }) +
        if
            (fun x =>
                match x with
                | 0 => 3
                | 1 => 0
                | 2 => 0) =
              b then
          { fst := 1, snd := 0 }
        else { fst := 0, snd := 0 }) -
      if
          (fun x =>
              match x with
              | 0 => 3
              | 1 => 1
              | 2 => 1) =
            b then
        { fst := 1, snd := 0 }
      else { fst := 0, snd := 0 }) =
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ ¬b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 } else 0
  decide +kernelAll goals completed! 🐙

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma smul_eq_self (Λ : SL(2,ℂ)) : Λ • pauliMatrix = pauliMatrix := byΛ:SL(2, ℂ)⊢ Λ • σ = σ
  rw [smul_eq,Λ:SL(2, ℂ)⊢ toTensor.symm (Λ • toTensor σ) = σΛ:SL(2, ℂ)⊢ toTensor.symm (toTensor σ) = σ toTensor_eq_asConsTensor,Λ:SL(2, ℂ)⊢ toTensor.symm (Λ • fromConstTriple asConsTensor) = σΛ:SL(2, ℂ)⊢ toTensor.symm (toTensor σ) = σ actionT_fromConstTriple,Λ:SL(2, ℂ)⊢ toTensor.symm (fromConstTriple asConsTensor) = σΛ:SL(2, ℂ)⊢ toTensor.symm (toTensor σ) = σ ← toTensor_eq_asConsTensorΛ:SL(2, ℂ)⊢ toTensor.symm (toTensor σ) = σΛ:SL(2, ℂ)⊢ toTensor.symm (toTensor σ) = σ]Λ:SL(2, ℂ)⊢ toTensor.symm (toTensor σ) = σ
  simpAll goals completed! 🐙

set_option backward.isDefEq.respectTransparency false in
@[simp]
lemma toTensor_smul_eq_self (Λ : SL(2,ℂ)) : Λ • σ^^^ = σ^^^ := byΛ:SL(2, ℂ)⊢ Λ • toTensor σ = toTensor σ
  rw [toTensor_eq_asConsTensorΛ:SL(2, ℂ)⊢ Λ • fromConstTriple asConsTensor = fromConstTriple asConsTensorΛ:SL(2, ℂ)⊢ Λ • fromConstTriple asConsTensor = fromConstTriple asConsTensor]Λ:SL(2, ℂ)⊢ Λ • fromConstTriple asConsTensor = fromConstTriple asConsTensor
  simpAll goals completed! 🐙

Variations of the pauli tensor

The Pauli matrices as the complex Lorentz tensor σ_μ^α^{dot β}.

abbrev pauliCo : ℂT[.down, .upL, .upR] :=
  permT id (IsReindexing.auto) {η' | μ ν ⊗ σ^^^ | ν α β}ᵀ

@[inherit_doc pauliCo]
scoped[PauliMatrix] notation "σ_^^" => PauliMatrix.pauliCo

The Pauli matrices as the complex Lorentz tensor σ_μ_{dot β}_α.

abbrev pauliCoDown : ℂT[.down, .downR, .downL] :=
  permT id (IsReindexing.auto) {σ_^^ | μ α β ⊗ εR' | β β' ⊗ εL' | α α' }ᵀ

@[inherit_doc pauliCoDown]
scoped[PauliMatrix] notation "σ___" => PauliMatrix.pauliCoDown

The Pauli matrices as the complex Lorentz tensor σ^μ_{dot β}_α.

abbrev pauliContrDown : ℂT[.up, .downR, .downL] :=
    permT id (IsReindexing.auto) {σ^^^ | μ α β ⊗ εR' | β β' ⊗ εL' | α α'}ᵀ

@[inherit_doc pauliContrDown]
scoped[PauliMatrix] notation "σ^__" => PauliMatrix.pauliContrDown

Different forms

              

lemma pauliCo_eq_ofRat : pauliCo = ofRat (fun b =>
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 4 = repDim (![Color.down, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (0 : Fin 4) ∧ b 1 = b 2 then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 4 = repDim (![Color.down, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (1 : Fin 4) ∧ b 1 ≠ b 2 then ⟨-1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 4 = repDim (![Color.down, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨0, 1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 4 = repDim (![Color.down, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨0, -1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 4 = repDim (![Color.down, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨-1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 4 = repDim (![Color.down, Color.upL, Color.upR] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ 2 = repDim (![Color.down, Color.upL, Color.upR] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨1, 0⟩ else ⟨0, 0⟩) := by⊢ σ_^^ =
  ofRat fun b =>
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis ![Color.down, Color.upL, Color.upR]).repr σ_^^ =
  (Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
    (ofRat fun b =>
      if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 })
  ext bb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr σ_^^) b =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [pauliCob:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      ((permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ)))))
    b =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      ((permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ)))))
    b =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      ((permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ)))))
    b =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [permT_basis_repr_symm_applyb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ (((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 1 2)).repr
      ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ))))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i))) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ (((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 1 2)).repr
      ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ))))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i))) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ (((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 1 2)).repr
      ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ))))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i))) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [contrT_basis_repr_applyb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ∑ b',
    ((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR])).repr
          ((prodT η') (toTensor σ)))
        ↑b' *
      (complexLorentzTensor.contr (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1))
        ((match Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match complexLorentzTensor.τ (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 2))) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ∑ b',
    ((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR])).repr
          ((prodT η') (toTensor σ)))
        ↑b' *
      (complexLorentzTensor.contr (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1))
        ((match Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match complexLorentzTensor.τ (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 2))) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ ∑ b',
    ((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR])).repr
          ((prodT η') (toTensor σ)))
        ↑b' *
      (complexLorentzTensor.contr (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1))
        ((match Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match complexLorentzTensor.τ (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 2))) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  conv_lhs =>
    enter [2, x]b:ComponentIdx ![Color.down, Color.upL, Color.upR]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i)))| ((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR])).repr ((prodT η') (toTensor σ)))
    ↑x *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1))
    ((match Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        (↑x 1) ⊗ₜ[ℂ]
      (match complexLorentzTensor.τ (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] 1) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 2)))
    rw [contr_basis_ratComplexNum]b:ComponentIdx ![Color.down, Color.upL, Color.upR]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i)))| ((Tensor.basis (Fin.append ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR])).repr ((prodT η') (toTensor σ)))
    ↑x *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0)
    rw [prodT_basis_repr_apply]b:ComponentIdx ![Color.down, Color.upL, Color.upR]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i)))| ((Tensor.basis ![Color.down, Color.down]).repr η') (ComponentIdx.prod ↑x).1 *
    ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr (toTensor σ)) (ComponentIdx.prod ↑x).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0)
    simp only [coMetric_eq_ofRat, ofRat_basis_repr_apply, toTensor_eq_ofRat]b:ComponentIdx ![Color.down, Color.upL, Color.upR]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i)))| Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
        1
      else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
    Physlib.RatComplexNum.toComplexNum
      (if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                  { fst := -1, snd := 0 }
                else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.down, Color.upL, Color.upR]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i)))| Physlib.RatComplexNum.toComplexNum
    ((if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
        1
      else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
      if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                  { fst := -1, snd := 0 }
                else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.down, Color.upL, Color.upR]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCo._proof_2 i)))| Physlib.RatComplexNum.toComplexNum
  (((if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
        1
      else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
      if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                  { fst := -1, snd := 0 }
                else 0) *
    if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0)
  rw [← map_sum Physlib.RatComplexNum.toComplexNumb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [ofRat_basis_repr_applyb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 })b:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 })]b:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 })
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx ![Color.down, Color.upL, Color.upR]⊢ (∑ x,
    ((if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
          1
        else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
        if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                    { fst := -1, snd := 0 }
                  else 0) *
      if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
  if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
  else
    if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }
  revert b⊢ ∀ (b : ComponentIdx ![Color.down, Color.upL, Color.upR]),
  (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod ↑x).1 0 = (ComponentIdx.prod ↑x).1 1 then -1 else 0) *
          if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x).2 2 = Fin.cast ⋯ 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 2)) then 1 else 0) =
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }
  decide +kernelAll goals completed! 🐙

lemma pauliCoDown_eq_ofRat : pauliCoDown = ofRat (fun b =>
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 4 = repDim (![Color.down, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (0 : Fin 4) ∧ b 1 = b 2 then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 4 = repDim (![Color.down, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (1 : Fin 4) ∧ b 1 ≠ b 2 then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 4 = repDim (![Color.down, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨0, -1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 4 = repDim (![Color.down, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨0, 1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 4 = repDim (![Color.down, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨-1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 4 = repDim (![Color.down, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ 2 = repDim (![Color.down, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨1, 0⟩ else ⟨0, 0⟩) := by⊢ σ___ =
  ofRat fun b =>
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis ![Color.down, Color.downR, Color.downL]).repr σ___ =
  (Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
    (ofRat fun b =>
      if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 })
  ext bb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr σ___) b =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [pauliCoDownb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      ((permT id pauliCoDown._proof_1)
        ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))))
    b =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      ((permT id pauliCoDown._proof_1)
        ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))))
    b =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      ((permT id pauliCoDown._proof_1)
        ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))))
    b =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [permT_basis_repr_symm_applyb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ (((Tensor.basis
          (Fin.append
              (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] ∘
            Fin.succSuccAbove 1 3)).repr
      ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL')))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i))) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ (((Tensor.basis
          (Fin.append
              (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] ∘
            Fin.succSuccAbove 1 3)).repr
      ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL')))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i))) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ (((Tensor.basis
          (Fin.append
              (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] ∘
            Fin.succSuccAbove 1 3)).repr
      ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL')))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i))) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [contrT_basis_repr_applyb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ∑ b',
    ((Tensor.basis
              (Fin.append
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL])).repr
          ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))
        ↑b' *
      (complexLorentzTensor.contr
          (Fin.append
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1))
        ((match
              Fin.append
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append
                  (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                  ![Color.downL, Color.downL] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ∑ b',
    ((Tensor.basis
              (Fin.append
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL])).repr
          ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))
        ↑b' *
      (complexLorentzTensor.contr
          (Fin.append
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1))
        ((match
              Fin.append
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append
                  (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                  ![Color.downL, Color.downL] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ ∑ b',
    ((Tensor.basis
              (Fin.append
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL])).repr
          ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))
        ↑b' *
      (complexLorentzTensor.contr
          (Fin.append
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1))
        ((match
              Fin.append
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append
                  (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                  ![Color.downL, Color.downL] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  conv_lhs =>
    enter [2, x]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| ((Tensor.basis
          (Fin.append
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL])).repr
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))
    ↑x *
  (complexLorentzTensor.contr
      (Fin.append (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
        ![Color.downL, Color.downL] 1))
    ((match
          Fin.append
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        (↑x 1) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append
              (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] 1) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    rw [contr_basis_ratComplexNum]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| ((Tensor.basis
          (Fin.append
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL])).repr
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL'))
    ↑x *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [prodT_basis_repr_apply]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| ((Tensor.basis
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)).repr
        ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR')))
      (ComponentIdx.prod ↑x).1 *
    ((Tensor.basis ![Color.downL, Color.downL]).repr εL') (ComponentIdx.prod ↑x).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [contrT_basis_repr_apply]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| (∑ b',
      ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr
            ((prodT σ_^^) εR'))
          ↑b' *
        (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2))
          ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2 with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              (↑b' 2) ⊗ₜ[ℂ]
            (match
                complexLorentzTensor.τ
                  (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2) with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              ((basisIdxCongr ⋯) (↑b' 3)))) *
    ((Tensor.basis ![Color.downL, Color.downL]).repr εL') (ComponentIdx.prod ↑x).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    simp only [coMetric_eq_ofRat, ofRat_basis_repr_apply,
      dualLeftMetric_eq_ofRat]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| (∑ b',
      ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr
            ((prodT σ_^^) εR'))
          ↑b' *
        (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2))
          ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2 with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              (↑b' 2) ⊗ₜ[ℂ]
            (match
                complexLorentzTensor.τ
                  (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2) with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              ((basisIdxCongr ⋯) (↑b' 3)))) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    enter [1, 1, 2, y]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr ((prodT σ_^^) εR'))
    ↑y *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2))
    ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        (↑y 2) ⊗ₜ[ℂ]
      (match complexLorentzTensor.τ (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR] 2) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑y 3)))
    rw [contr_basis_ratComplexNum]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr ((prodT σ_^^) εR'))
    ↑y *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    rw [prodT_basis_repr_apply]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr σ_^^) (ComponentIdx.prod ↑y).1 *
    ((Tensor.basis ![Color.downR, Color.downR]).repr εR') (ComponentIdx.prod ↑y).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    simp only [coMetric_eq_ofRat, ofRat_basis_repr_apply, pauliCo_eq_ofRat,
      dualRightMetric_eq_ofRat]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| Physlib.RatComplexNum.toComplexNum
      (if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| Physlib.RatComplexNum.toComplexNum
    ((if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
      if
          (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| Physlib.RatComplexNum.toComplexNum
  (((if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
      if
          (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
    if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
  conv_lhs =>
    enter [2, x]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| (∑ y,
      Physlib.RatComplexNum.toComplexNum
        (((if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) *
            if
                (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← map_sum Physlib.RatComplexNum.toComplexNum]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| Physlib.RatComplexNum.toComplexNum
      (∑ x_1,
        ((if
                (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                  (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) *
            if
                (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| Physlib.RatComplexNum.toComplexNum
    ((∑ x_1,
        ((if
                (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                  (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) *
            if
                (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.down, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliCoDown._proof_1 i)))| Physlib.RatComplexNum.toComplexNum
  (((∑ x_1,
        ((if
                (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                  (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) *
            if
                (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
    if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
  rw [← map_sum Physlib.RatComplexNum.toComplexNumb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    bb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b]b:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ((Tensor.basis ![Color.down, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }))
    b
  rw [ofRat_basis_repr_applyb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 })b:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 })]b:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 })
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx ![Color.down, Color.downR, Color.downL]⊢ (∑ x,
    ((∑ x_1,
          ((if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                    (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                      (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := -1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                          { fst := 1, snd := 0 }
                        else { fst := 0, snd := 0 }) *
              if
                  (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                1
              else
                if
                    (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                  -1
                else 0) *
            if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
          1
        else
          if
              (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
            -1
          else 0) *
      if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
  else
    if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
            else { fst := 0, snd := 0 }
  revert b⊢ ∀ (b : ComponentIdx ![Color.down, Color.downR, Color.downL]),
  (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := -1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := 1, snd := 0 }
                          else { fst := 0, snd := 0 }) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := -1, snd := 0 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }
  decide +kernelAll goals completed! 🐙

lemma pauliContrDown_ofRat : pauliContrDown = ofRat (fun b =>
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 4 = repDim (![Color.up, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (0 : Fin 4) ∧ b 1 = b 2 then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 4 = repDim (![Color.up, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (1 : Fin 4) ∧ b 1 ≠ b 2 then ⟨-1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 4 = repDim (![Color.up, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨0, 1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 4 = repDim (![Color.up, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (2 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨0, -1⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 4 = repDim (![Color.up, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (1 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (1 : Fin 2) then ⟨1, 0⟩ else
    if b 0 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 4 = repDim (![Color.up, Color.downR, Color.downL] 0) rflAll goals completed! 🐙) (3 : Fin 4) ∧ b 1 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 1) rflAll goals completed! 🐙) (0 : Fin 2) ∧
      b 2 = Fin.cast (byb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ 2 = repDim (![Color.up, Color.downR, Color.downL] 2) rflAll goals completed! 🐙) (0 : Fin 2) then ⟨-1, 0⟩ else 0) := by⊢ σ^__ =
  ofRat fun b =>
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis ![Color.up, Color.downR, Color.downL]).repr σ^__ =
  (Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
    (ofRat fun b =>
      if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
            else
              if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
              else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0)
  ext bb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr σ^__) b =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b
  rw [pauliContrDownb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      ((permT id pauliContrDown._proof_1)
        ((contrT 3 1 3 pauliContrDown._proof_2)
          ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))))
    b =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      ((permT id pauliContrDown._proof_1)
        ((contrT 3 1 3 pauliContrDown._proof_2)
          ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))))
    b =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b]b:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      ((permT id pauliContrDown._proof_1)
        ((contrT 3 1 3 pauliContrDown._proof_2)
          ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))))
    b =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b
  rw [permT_basis_repr_symm_applyb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ (((Tensor.basis
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] ∘
            Fin.succSuccAbove 1 3)).repr
      ((contrT 3 1 3 pauliContrDown._proof_2)
        ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL')))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i))) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ (((Tensor.basis
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] ∘
            Fin.succSuccAbove 1 3)).repr
      ((contrT 3 1 3 pauliContrDown._proof_2)
        ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL')))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i))) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b]b:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ (((Tensor.basis
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] ∘
            Fin.succSuccAbove 1 3)).repr
      ((contrT 3 1 3 pauliContrDown._proof_2)
        ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL')))
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i))) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b
  rw [contrT_basis_repr_applyb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ∑ b',
    ((Tensor.basis
              (Fin.append
                (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL])).repr
          ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))
        ↑b' *
      (complexLorentzTensor.contr
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1))
        ((match
              Fin.append
                (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append
                  (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                  ![Color.downL, Color.downL] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ∑ b',
    ((Tensor.basis
              (Fin.append
                (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL])).repr
          ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))
        ↑b' *
      (complexLorentzTensor.contr
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1))
        ((match
              Fin.append
                (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append
                  (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                  ![Color.downL, Color.downL] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b]b:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ ∑ b',
    ((Tensor.basis
              (Fin.append
                (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL])).repr
          ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))
        ↑b' *
      (complexLorentzTensor.contr
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1))
        ((match
              Fin.append
                (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                ![Color.downL, Color.downL] 1 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            (↑b' 1) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append
                  (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
                  ![Color.downL, Color.downL] 1) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => complexContrBasisFin4
            | Color.down => complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b
  conv_lhs =>
    enter [2, x]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| ((Tensor.basis
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL])).repr
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))
    ↑x *
  (complexLorentzTensor.contr
      (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
        ![Color.downL, Color.downL] 1))
    ((match
          Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL] 1 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        (↑x 1) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append
              (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
              ![Color.downL, Color.downL] 1) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    rw [contr_basis_ratComplexNum]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| ((Tensor.basis
          (Fin.append (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)
            ![Color.downL, Color.downL])).repr
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL'))
    ↑x *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [prodT_basis_repr_apply]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| ((Tensor.basis (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] ∘ Fin.succSuccAbove 2 3)).repr
        ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR')))
      (ComponentIdx.prod ↑x).1 *
    ((Tensor.basis ![Color.downL, Color.downL]).repr εL') (ComponentIdx.prod ↑x).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [contrT_basis_repr_apply]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| (∑ b',
      ((Tensor.basis (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr
            ((prodT (toTensor σ)) εR'))
          ↑b' *
        (complexLorentzTensor.contr (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2))
          ((match Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2 with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              (↑b' 2) ⊗ₜ[ℂ]
            (match
                complexLorentzTensor.τ (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2) with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              ((basisIdxCongr ⋯) (↑b' 3)))) *
    ((Tensor.basis ![Color.downL, Color.downL]).repr εL') (ComponentIdx.prod ↑x).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    simp only [coMetric_eq_ofRat, ofRat_basis_repr_apply,
      dualLeftMetric_eq_ofRat]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| (∑ b',
      ((Tensor.basis (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr
            ((prodT (toTensor σ)) εR'))
          ↑b' *
        (complexLorentzTensor.contr (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2))
          ((match Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2 with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              (↑b' 2) ⊗ₜ[ℂ]
            (match
                complexLorentzTensor.τ (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2) with
              | Color.upL => LeftHandedWeyl.basis
              | Color.downL => DualLeftHandedWeyl.basis
              | Color.upR => RightHandedWeyl.basis
              | Color.downR => DualRightHandedWeyl.basis
              | Color.up => complexContrBasisFin4
              | Color.down => complexCoBasisFin4)
              ((basisIdxCongr ⋯) (↑b' 3)))) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    enter [1, 1, 2, y]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| ((Tensor.basis (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr
      ((prodT (toTensor σ)) εR'))
    ↑y *
  (complexLorentzTensor.contr (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2))
    ((match Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        (↑y 2) ⊗ₜ[ℂ]
      (match complexLorentzTensor.τ (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR] 2) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => complexContrBasisFin4
        | Color.down => complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑y 3)))
    rw [contr_basis_ratComplexNum]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| ((Tensor.basis (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.downR, Color.downR])).repr
      ((prodT (toTensor σ)) εR'))
    ↑y *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    rw [prodT_basis_repr_apply]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr (toTensor σ)) (ComponentIdx.prod ↑y).1 *
    ((Tensor.basis ![Color.downR, Color.downR]).repr εR') (ComponentIdx.prod ↑y).2 *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    simp only [coMetric_eq_ofRat,ofRat_basis_repr_apply, toTensor_eq_ofRat,
      dualRightMetric_eq_ofRat]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| Physlib.RatComplexNum.toComplexNum
      (if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := -1, snd := 0 }
                else 0) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| Physlib.RatComplexNum.toComplexNum
    ((if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := -1, snd := 0 }
                else 0) *
      if
          (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))y:↥(ComponentIdx.prod ↑x).1.DropPairSection| Physlib.RatComplexNum.toComplexNum
  (((if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
        { fst := 1, snd := 0 }
      else
        if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                  (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                    (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := -1, snd := 0 }
                else 0) *
      if
          (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
    if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)
  conv_lhs =>
    enter [2, x]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| (∑ y,
      Physlib.RatComplexNum.toComplexNum
        (((if (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 1 = (ComponentIdx.prod ↑y).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑y).1 1 ≠ (ComponentIdx.prod ↑y).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑y).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑y).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑y).1 2 = Fin.cast ⋯ 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑y).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑y).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑y 2) = ↑((basisIdxCongr ⋯) (↑y 3)) then 1 else 0)) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← map_sum Physlib.RatComplexNum.toComplexNum]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| Physlib.RatComplexNum.toComplexNum
      (∑ x_1,
        ((if
                (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                  (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| Physlib.RatComplexNum.toComplexNum
    ((∑ x_1,
        ((if
                (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                  (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx ![Color.up, Color.downR, Color.downL]x:↥(ComponentIdx.DropPairSection fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv id pauliContrDown._proof_1 i)))| Physlib.RatComplexNum.toComplexNum
  (((∑ x_1,
        ((if
                (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                  (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                    (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
              1
            else
              if
                  (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                -1
              else 0) *
          if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
        1
      else
        if
            (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
          -1
        else 0) *
    if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
  rw [← map_sum Physlib.RatComplexNum.toComplexNumb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    bb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b]b:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ((Tensor.basis ![Color.up, Color.downR, Color.downL]).repr
      (ofRat fun b =>
        if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
          else
            if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
            else
              if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
              else
                if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
                else
                  if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0))
    b
  rw [ofRat_basis_repr_applyb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0)b:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0)]b:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0)
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx ![Color.up, Color.downR, Color.downL]⊢ (∑ x,
    ((∑ x_1,
          ((if
                  (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                    (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                      (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                        (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                1
              else
                if
                    (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                  -1
                else 0) *
            if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
          1
        else
          if
              (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
            -1
          else 0) *
      if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
  else
    if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
        else
          if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
          else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0
  revert b⊢ ∀ (b : ComponentIdx ![Color.up, Color.downR, Color.downL]),
  (∑ x,
      ((∑ x_1,
            ((if
                    (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 0 ∧
                      (ComponentIdx.prod ↑x_1).1 1 = (ComponentIdx.prod ↑x_1).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 1 ∧
                        (ComponentIdx.prod ↑x_1).1 1 ≠ (ComponentIdx.prod ↑x_1).1 2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                          (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧ (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 2 ∧
                            (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                              (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                              (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 0 ∧
                                (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑x_1).1 0 = Fin.cast ⋯ 3 ∧
                                (ComponentIdx.prod ↑x_1).1 1 = Fin.cast ⋯ 1 ∧
                                  (ComponentIdx.prod ↑x_1).1 2 = Fin.cast ⋯ 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 1 then
                  1
                else
                  if
                      (ComponentIdx.prod ↑x_1).2 1 = Fin.cast dualRightMetric_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x_1).2 0 = Fin.cast dualRightMetric_eq_ofRat._proof_1 1 then
                    -1
                  else 0) *
              if ↑(↑x_1 2) = ↑((basisIdxCongr ⋯) (↑x_1 3)) then 1 else 0) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 1 then
            1
          else
            if
                (ComponentIdx.prod ↑x).2 1 = Fin.cast dualLeftMetric_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast dualLeftMetric_eq_ofRat._proof_1 1 then
              -1
            else 0) *
        if ↑(↑x 1) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
    if b 0 = Fin.cast ⋯ 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
    else
      if b 0 = Fin.cast ⋯ 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
      else
        if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 0, snd := 1 }
        else
          if b 0 = Fin.cast ⋯ 2 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 0 then { fst := 0, snd := -1 }
          else
            if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 1 ∧ b 2 = Fin.cast ⋯ 1 then { fst := 1, snd := 0 }
            else if b 0 = Fin.cast ⋯ 3 ∧ b 1 = Fin.cast ⋯ 0 ∧ b 2 = Fin.cast ⋯ 0 then { fst := -1, snd := 0 } else 0
  decide +kernelAll goals completed! 🐙

Group actions

The tensor pauliCo is invariant under the action of SL(2,ℂ).

set_option backward.isDefEq.respectTransparency false in


lemma smul_pauliCo (g : SL(2,ℂ)) : g • pauliCo = pauliCo := byg:SL(2, ℂ)⊢ g • σ_^^ = σ_^^
  rw [← permT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) (g • (contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ))) = σ_^^g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT (g • η')) (g • toTensor σ))) = σ_^^ ← contrT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) (g • (prodT η') (toTensor σ))) = σ_^^g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT (g • η')) (g • toTensor σ))) = σ_^^ ← prodT_equivariantg:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT (g • η')) (g • toTensor σ))) = σ_^^g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT (g • η')) (g • toTensor σ))) = σ_^^]g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT (g • η')) (g • toTensor σ))) = σ_^^
  rw [toTensor_smul_eq_self,g:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT (g • η')) (toTensor σ))) = σ_^^All goals completed! 🐙 actionT_coMetricg:SL(2, ℂ)⊢ (permT id pauliCo._proof_2) ((contrT 3 1 2 pauliCo._proof_3) ((prodT η') (toTensor σ))) = σ_^^All goals completed! 🐙]All goals completed! 🐙

The tensor pauliCoDown is invariant under the action of SL(2,ℂ).

set_option backward.isDefEq.respectTransparency false in
set_option maxRecDepth 2000 in


lemma smul_pauliCoDown (g : SL(2,ℂ)) : g • pauliCoDown = pauliCoDown := byg:SL(2, ℂ)⊢ g • σ___ = σ___
  rw [← permT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    (g • (contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL')) =
  σ___g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___ ← contrT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2) (g • (prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL')) =
  σ___g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___ ← prodT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT (g • (contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) (g • εL'))) =
  σ___g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___
    ← contrT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) (g • (prodT σ_^^) εR'))) (g • εL'))) =
  σ___g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___ ← prodT_equivariantg:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___]g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT (g • σ_^^)) (g • εR')))) (g • εL'))) =
  σ___
  rw [smul_pauliCo,g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) (g • εR')))) (g • εL'))) =
  σ___All goals completed! 🐙 actionT_dualLeftMetric,g:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) (g • εR')))) εL')) =
  σ___All goals completed! 🐙 actionT_dualRightMetricg:SL(2, ℂ)⊢ (permT id pauliCoDown._proof_1)
    ((contrT 3 1 3 pauliCoDown._proof_2) ((prodT ((contrT 3 2 3 pauliCoDown._proof_4) ((prodT σ_^^) εR'))) εL')) =
  σ___All goals completed! 🐙]All goals completed! 🐙

The tensor pauliContrDown is invariant under the action of SL(2,ℂ).

set_option backward.isDefEq.respectTransparency false in


lemma smul_pauliContrDown (g : SL(2,ℂ)) : g • pauliContrDown = pauliContrDown := byg:SL(2, ℂ)⊢ g • σ^__ = σ^__
  rw [← permT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    (g •
      (contrT 3 1 3 pauliContrDown._proof_2)
        ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL')) =
  σ^__g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__ ← contrT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      (g • (prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL')) =
  σ^__g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__ ← prodT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT (g • (contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) (g • εL'))) =
  σ^__g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__
    ← contrT_equivariant,g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) (g • (prodT (toTensor σ)) εR'))) (g • εL'))) =
  σ^__g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__ ← prodT_equivariantg:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__]g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (g • toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__
  rw [toTensor_smul_eq_self,g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) (g • εR')))) (g • εL'))) =
  σ^__All goals completed! 🐙 actionT_dualLeftMetric,g:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) (g • εR')))) εL')) =
  σ^__All goals completed! 🐙 actionT_dualRightMetricg:SL(2, ℂ)⊢ (permT id pauliContrDown._proof_1)
    ((contrT 3 1 3 pauliContrDown._proof_2)
      ((prodT ((contrT 3 2 3 pauliContrDown._proof_4) ((prodT (toTensor σ)) εR'))) εL')) =
  σ^__All goals completed! 🐙]All goals completed! 🐙